C(q) is the least amount of money needed to buy inputs that will produce output q. It is the cost function.
FC are fixed costs, the costs incurred even if there is no production. FC = C(0).
VC(q) are variable costs. VC(q) = C(q) - FC.
AC(q) is average cost. AC(q) = C(q)/q.
AVC is average variable cost. AC(q) = VC(q)/q.
AFC is average fixed cost. AFC(q) = FC/q. AFC(0) is infinite and AFC(inf.) is zero. (yes, I should have said that the limit as q approaches infinity of AFC is zero, but I am sure you can tolerate the short hand.)
MC(q) is marginal cost. It is the cost of making the next unit. MC(q) is approximately C(q+1) - C(q). Put the other way, C(q+1) is approximately C(q) + MC(q). The cost of making q+1 units is the cost of making q units plus marginal cost at q. A Diagram.
Assuming that FC are not zero, AC(0) will be infinite and AC(inf) = AVC(inf)
Whenever AC is increasing, MC is above AC. When decreasing MC is below AC.
AC(q+1) - AC(q) = c(q+1)/(q+1) - c(q)/q = c(q)/(q+1) + mc(q)/(q+1) - c(q)/q
= -c(q)/(q(q+1) + mc(q)/q+1)
= (1/(q+1)) (mc(q) - ac(q) ) so mc greater than ac means ac(q+1) > ac(q) or ac increasing.
This means that MC goes through the minimum point of AC. Note that MC = VC(q+1) - VC(q). (why?) so the above proof also shows that MC goes through the minimum point of AVC.
Firms maximize profits. p = p q - c(q) = q (p - ac(q) ).
A necessary condition for maximizing profits is that p = mc(q). We assume mc slopes up.
Part of a proof: We will show that if p = mc(q*) then profits go down if one decreases q* by one unit. One can show, but we will not, that increasing q above q* will also decrease profits. Thus q* is a local profit maximum. Our claim is that
p(q*) - p(q*-1) is positive. Now p(q*) - p(q*-1) =
[pq* - c(q*)] - [ p (q*-1) - c(q*-1)]
= p - [ c(q*) - c(q*-1) ]
= p - mc(q*-1)
which is positive because mc slopes up and equals p at q*. (draw the picture)
(at home construct the same argument for for q*+1 and q*+2. do you see the problem with the discrete approximation of mc as c(q*+1) - c(q*) rather than the calculus answer of mc is limt -> 0 [c(q+t) - c(q)]/t.)
The shutdown point is given by p = minimum point on avc. When p = min avc = mc , profits will be exactly - FC. p = q ( p - ac) = q ( p - avc) - fc but p = avc so p = -fc. Profits would be less if p were less so p less than min avc = mc, best policy is shutdown.
The supply curve is mc above avc.